Average Error: 37.4 → 25.3
Time: 18.0s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -7.694926734883939 \cdot 10^{-264}:\\ \;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}} \cdot 0.5\\ \mathbf{elif}\;re \le 3.185198141526603 \cdot 10^{+94}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(re + \sqrt{\sqrt[3]{im \cdot im + re \cdot re} \cdot \left(\sqrt[3]{im \cdot im + re \cdot re} \cdot \sqrt[3]{im \cdot im + re \cdot re}\right)}\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(re + re\right)} \cdot 0.5\\ \end{array}\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -7.694926734883939 \cdot 10^{-264}:\\
\;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}} \cdot 0.5\\

\mathbf{elif}\;re \le 3.185198141526603 \cdot 10^{+94}:\\
\;\;\;\;\sqrt{2.0 \cdot \left(re + \sqrt{\sqrt[3]{im \cdot im + re \cdot re} \cdot \left(\sqrt[3]{im \cdot im + re \cdot re} \cdot \sqrt[3]{im \cdot im + re \cdot re}\right)}\right)} \cdot 0.5\\

\mathbf{else}:\\
\;\;\;\;\sqrt{2.0 \cdot \left(re + re\right)} \cdot 0.5\\

\end{array}
double f(double re, double im) {
        double r8703829 = 0.5;
        double r8703830 = 2.0;
        double r8703831 = re;
        double r8703832 = r8703831 * r8703831;
        double r8703833 = im;
        double r8703834 = r8703833 * r8703833;
        double r8703835 = r8703832 + r8703834;
        double r8703836 = sqrt(r8703835);
        double r8703837 = r8703836 + r8703831;
        double r8703838 = r8703830 * r8703837;
        double r8703839 = sqrt(r8703838);
        double r8703840 = r8703829 * r8703839;
        return r8703840;
}
↓
double f(double re, double im) {
        double r8703841 = re;
        double r8703842 = -7.694926734883939e-264;
        bool r8703843 = r8703841 <= r8703842;
        double r8703844 = im;
        double r8703845 = r8703844 * r8703844;
        double r8703846 = 2.0;
        double r8703847 = r8703845 * r8703846;
        double r8703848 = sqrt(r8703847);
        double r8703849 = r8703841 * r8703841;
        double r8703850 = r8703845 + r8703849;
        double r8703851 = sqrt(r8703850);
        double r8703852 = r8703851 - r8703841;
        double r8703853 = sqrt(r8703852);
        double r8703854 = r8703848 / r8703853;
        double r8703855 = 0.5;
        double r8703856 = r8703854 * r8703855;
        double r8703857 = 3.185198141526603e+94;
        bool r8703858 = r8703841 <= r8703857;
        double r8703859 = cbrt(r8703850);
        double r8703860 = r8703859 * r8703859;
        double r8703861 = r8703859 * r8703860;
        double r8703862 = sqrt(r8703861);
        double r8703863 = r8703841 + r8703862;
        double r8703864 = r8703846 * r8703863;
        double r8703865 = sqrt(r8703864);
        double r8703866 = r8703865 * r8703855;
        double r8703867 = r8703841 + r8703841;
        double r8703868 = r8703846 * r8703867;
        double r8703869 = sqrt(r8703868);
        double r8703870 = r8703869 * r8703855;
        double r8703871 = r8703858 ? r8703866 : r8703870;
        double r8703872 = r8703843 ? r8703856 : r8703871;
        return r8703872;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.4
Target32.3
Herbie25.3
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if re < -7.694926734883939e-264

    1. Initial program 46.1

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
    2. Using strategy rm

    3. Applied flip-+46.0

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} - re}}}\]

    4. Applied associate-*r/46.1

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re\right)}{\sqrt{re \cdot re + im \cdot im} - re}}}\]

    5. Applied sqrt-div46.1

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\]

    6. Simplified34.0

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2.0 \cdot \left(im \cdot im\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]

    if -7.694926734883939e-264 < re < 3.185198141526603e+94

    1. Initial program 21.3

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt21.5

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\color{blue}{\left(\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}\right) \cdot \sqrt[3]{re \cdot re + im \cdot im}}} + re\right)}\]

    if 3.185198141526603e+94 < re

    1. Initial program 48.1

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]

    2. Taylor expanded around inf 9.3

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{re} + re\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification25.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -7.694926734883939 \cdot 10^{-264}:\\ \;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}} \cdot 0.5\\ \mathbf{elif}\;re \le 3.185198141526603 \cdot 10^{+94}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(re + \sqrt{\sqrt[3]{im \cdot im + re \cdot re} \cdot \left(\sqrt[3]{im \cdot im + re \cdot re} \cdot \sqrt[3]{im \cdot im + re \cdot re}\right)}\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(re + re\right)} \cdot 0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 
(FPCore (re im)
  :name "math.sqrt on complex, real part"

  :herbie-target
  (if (< re 0) (* 0.5 (* (sqrt 2) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))